Mathematics & Physics 2016, 9(3), 307–309 УДК 517.55 On Zeros of Holomorphic Functions Olga V.Khodos∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 29.02.2016, received in revised form 06.04.2016, accepted 16.05.2016 The aim of the article is to find conditions on the coefficients of the Taylor expansion of a holomorphic function in C that guarantee a absence of zeros. <...> The aim of the article is to find conditions on the coefficients of the Taylor expansion of a holomorphic function in C that guarantee an absence of zeros. <...> Let a function f = f(z) with respect to complex variable z be holomorphic in a neighborhood of zero in the complex plane C: f(z) = Let γr be a circle of the form γr = {z : |z| = r}, r > 0. <...> For function f to be an entire function of finite order of growth which has no zeros, it is necessary and sufficient that for sufficiently small r there exists k0 ∈ N such that ∫ 1 γr zk df f = 0 при всех k k0. (2) In this case the minimum k0 is equal to the order of function. <...> Recall that the entire function f(z) has a finite order (of growth) if there exists a positive number A such that f(z) = O(erA ) for |z| = R→+∞. <...> The infimum of such numbers A is called the order of function (see, e.g., [2, 3]). <...> Let the function f be a function of finite order of growth, which has no zeros in C then it is well known that it has the form: f(z) = eϕ(z), where ϕ(z) is a polynomial of some degree k0 (see, e.g., [2, Ch. 7, Sec. 1.5]). <...> Since f(z) is holomorphic function in a neighborhood of zero and f(0) ̸= 0 then values of f(z) lie in a neighborhood of f(0) and this ∗Olga_Khodos@mail.ru ⃝ Siberian Federal University. <...> All rights reserved c – 307 – ∑ k=0 ∞ bkzk, f(0) = b0 = 1. (1) Olga V.Khodos On Zeros of Holomorphic Functions neighborhood does not contain the point 0 for sufficiently small <...>